Geometry & Topology Monographs 13 (2008) 229–259 229

Homotopy algebra of open–closed strings

HIROSHIGE KAJIURA

JIM STASHEFF

This paper is a survey of our previous works on open–closed homotopy algebras,together with geometrical background, especially in terms of compactifications ofconfiguration spaces (one of Fred’s specialities) of Riemann surfaces, structures onloop spaces, etc. We newly present Merkulov’s geometric A1–structure [48] as aspecial example of an OCHA. We also recall the relation of open–closed homotopyalgebras to various aspects of deformation theory.

18G55; 81T18

Dedicated to Fred Cohen in honor of his 60th birthday

1 Introduction

Open–closed homotopy algebras (OCHAs) (Kajiura and Stasheff [37]) are inspiredby Zwiebach’s open–closed string field theory [62], which is presented in terms ofdecompositions of moduli spaces of the corresponding Riemann surfaces. The Riemannsurfaces are (respectively) spheres with (closed string) punctures and disks with (openstring) punctures on the boundaries. That is, from the viewpoint of conformal fieldtheory, classical closed string field theory is related to the conformal plane C withpunctures and classical open string field theory is related to the upper half planeH with punctures on the boundary. Thus classical closed string field theory has anL1–structure (Zwiebach [61], Stasheff [56], Kimura, Stasheff and Voronov [40]) andclassical open string field theory has an A1–structure (Gaberdiel and Zwiebach [13],Zwiebach [62], Nakatsu [50] and Kajiura [35]). The algebraic structure, we call it anOCHA, that the classical open–closed string field theory has is similarly interestingsince it is related to the upper half plane H with punctures both in the bulk and on theboundary.

In operad theory (see Markl, Shnider, and Stasheff [46]), the relevance of the littledisk operad to closed string theory is known, where a (little) disk is related to a closedstring puncture on a sphere in the Riemann surface picture above. The homology ofthe little disk operad defines a Gerstenhaber algebra (Cohen [7], Getzler and Jones

Published: 22 February 2008 DOI: 10.2140/gtm.2008.13.229

230 Hiroshige Kajiura and Jim Stasheff

[17]), in particular, a suitably compatible graded commutative algebra structure andgraded Lie algebra structure. The framed little disk operad is in addition equipped witha BV–operator which rotates the disk boundary S1 . The algebraic structure on thehomology is then a BV–algebra (Getzler [16]), where the graded commutative productand the graded Lie bracket are related by the BV–operator. Physically, closed stringstates associated to each disk boundary S1 are constrained to be the S1 –invariant parts,the kernel of the BV–operator. This in turn leads to concentrating on the Lie algebrastructure, where two disk boundaries are identified by twist-sewing as Zwiebach did[61]. On the other hand, he worked at the chain level (‘off shell’), discovering an L1–structure. This was important since the multi-variable operations of the L1–structureprovided correlators of closed string field theory. Similarly for open string theory, thelittle interval operad and associahedra are relevant, the homology corresponding toa graded associative algebra, but the chain level reveals an A1–structure giving thehigher order correlators of open string field theory.

The corresponding operad for the open–closed string theory is the Swiss-cheese operad(Voronov [60]) that combines the little disk operad with the little interval operad; itwas inspired also by Kontsevich’s approach to deformation quantization. The algebraicstructure at the homology level has been analyzed thoroughly by Harrelson [28]. Incontrast, our work in the open–closed case is at the level of strong homotopy algebra,combining the known but separate L1– and A1–structures.

In our earlier work, we defined such a homotopy algebra and called it an open–closedhomotopy algebra (OCHA) [37]. In particular, we showed that this description isa homotopy invariant algebraic structure, ie, that it transfers well under homotopyequivalences or quasi-isomorphisms. Also, we showed that an open–closed homotopyalgebra gives us a general scheme for deformation of open string structures (A1–algebras) by closed strings (L1–algebras).

In this paper, we aim to explain a background for OCHAs, the aspect of moduli spaces ofRiemann surfaces. Also, we present the relation of OCHAs with Merkulov’s geometricA1–structures [48; 49].

An open–closed homotopy algebra consists of a direct sum of graded vector spacesHDHc˚Ho . It has an L1–structure on Hc and reduces to an A1–algebra if weset Hc D 0. Moreover, the operations that intertwine the two are a generalization of thestrong homotopy analog of H Cartan’s notion of a Lie algebra g acting on a differentialgraded algebra E (Cartan [6], Flato, Gerstenhaber and Voronov [11]). In Section 2, westart from discussing the moduli space aspects and the associated operad (tree graph)structures for A1–algebras, L1–algebras, and then OCHAs, together with recallingother descriptions by multi-variable operations and coderivation differentials. In a more

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Homotopy algebra of open–closed strings 231

physically oriented paper [38], we gave an alternative interpretation in the language ofhomological vector fields on a supermanifold.

One of the key theorems in homotopy algebra is the minimal model theorem whichwas first proved for A1–algebras by Kadeishvili [34]. The minimal model theoremstates the existence of minimal models for homotopy algebras analogous to Sullivan’sminimal models [58] for differential graded commutative algebras introduced in thecontext of rational homotopy theory. In Section 3 we re-state the minimal modeltheorem for our open–closed homotopy algebras.

As suggested by Merkulov, his geometric A1–structure [48] is a special example ofan OCHA. In Section 4, we present a new formulation of an OCHA in Merkulov’sframework.

In Section 5, we recall the relation of open–closed homotopy algebras to various aspectsof deformation theory and the relevant moduli spaces and in Section 6, return to therelation to the motivating string theory.

There is a distinction between the historical grading used in defining A1– and L1–structures and the more recent one common in the physics literature. They are relatedby (de)suspension of the underlying graded vector spaces. Since we emphasize theversions in terms of a single differential of degree one on the relevant ‘standardconstruction’, we will only occassionally refer to the older version, primarily forungraded strictly associative or Lie algebras or strict differential graded algebras. Thedistinction does influence the exposition, but the only importance technically is thesigns that occur. However, the detailed signs are conceptually unimportant (althoughcrucial in calculations), so we indicate them here only with ˙, the precise details beingin [37; 38].

We restrict our arguments to the case that the characteristic of the field k is zero. Wefurther let k D C for simplicity.

2 Strong homotopy algebra

2.1 Topology of based loop spaces

An open–closed homotopy algebra [37] is a strong homotopy algebra (or 1–algebra)which combines two typical strong homotopy algebras, an A1–algebra and an L1–algebra.

An A1–algebra was introduced [53] as a structure exemplified by the chains on thebased loop space Y WD �X of a topological space X with a base point x0 2 X: A

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232 Hiroshige Kajiura and Jim Stasheff

based loop x 2 Y WD�X is a map xW Œ0; 1�! X such that x.0/D x.1/D x0 . Thebased loop space Y forms a group-like space, where the product

m2W Y �Y ! Y

is given naturally by connecting two based loops as usual. The product m2 is notassociative but there exists a homotopy between m2.m2�1/ and m2.1�m2/ describedby an interval K3 (Figure 1 (a))

m3W K3 �Y �Y �Y �! Y :

In a similar way, we can consider possible operations of .Y /�4! Y constructed fromm2 and m3 . These connect to form a map on the boundary of a pentagon, which canbe extended to a pentagon K4 (Figure 1 (b)), providing a higher homotopy operation:

m4W K4 �Y �4�! Y :

(• •) • • (• •)

(a) (b)

K3K4

Figure 1: (a) An interval as associahedra K3 (b) A pentagon asassociahedra K4

Repeating this procedure leads to higher dimensional polytopes Kn , of dimension.n� 2/ [53], now called associahedra since the vertices correspond to all ways ofassociating a string of n letters. For Y D�X , we have higher homotopies

mnW Kn � .Y /�n�! Y

extending maps on the boundary of Kn defined by compositions of the mk for k < n.Then, a topological space Y equipped with the structures fmn;Kngn�2 as above iscalled an A1–space.

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Homotopy algebra of open–closed strings 233

2.2 Compactification of moduli spaces of disks with boundary punctures

Although it was not noticed for many years, the associahedra Kn can be obtained asthe moduli space of the real compactification of the configuration space of .n� 2/

distinct points in an interval or to a real compactification MnC1 of the moduli spaceMnC1 of a disk with .nC 1/ points on the boundary (Figure 2 (a)).

(a) (b)

'

1 1

1 0

Figure 2: (a) The identification of the interval with .n�2/ points on it withthe boundary of the disk with .nC 1/ points on the boundary(b) The correspondence of the compactification of the moduli spaces withtree graphs; the case of a boundary component of the compactified modulispace M7

This compactification can be related directly to the planar tree operad (Figure 2 (b)).MnC1 is described as the configuration space of .nC1/–punctures on S1 �R[f1g

divided by conformal transformations. In the case in which the Riemann surface isthe disk, the conformal transformations form SL.2;R/, whose element g 2 SL.2;R/

acts on R[f1g as

(1) g.x/DaxC b

cxC d; x 2 .R[f1g/ ; g WD

�a b

c d

�2 SL.2;R/ :

This degree of freedom can be killed by fixing three points on the boundary. Usuallywe set the three points at 0, 1 and 1. We take the point 1 as the ‘root edge’.

Then, the interval is identified with the arc between 0 and 1 as in Figure 2 (a). Thus,we obtain:

(2) MnC1 D f.t2; : : : ; tn�1/ j 0< t2 < t3 < � � �< tn�1 < 1g :

The real compactifications MnC1 of Axelrod and Singer [1] (the real analog of theFulton–MacPherson compactification [12]), that is, the compactifications of MnC1

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234 Hiroshige Kajiura and Jim Stasheff

with real codimension one boundaries, are in fact combinatorially homeomorphic tothe Stasheff associahedron Kn . For instance, we rather obviously have:

ı for nD 2, M2C1 ' fptg ' xM2C1 'K2 ,

ı for nD 3, M3C1 ' ft2 j 0< t2 < 1g and xM3C1 'K3 ' the closed interval.

For n> 3, the particulars of the compactification process account for the compactifica-tion being combinatorially homeomorphic to K4 rather than to the closed simplex.

2.3 Tree formulation

There are some advantages to indexing the maps mk and their compositions by planarrooted trees (as originally suggested by Frank Adams around 1960, when trees wouldhave had to be inserted in manuscripts by hand); e.g. mk will correspond to the corollaMk with k leaves all attached directly to the root. The composite mk �i ml thencorresponds to grafting the root of Ml to the i -th leaf of Mk , reading from left toright (see Figure 3). (Thus �i is a precise analog of Gerstenhaber’s ıi , although thecorrespondence was not observed for a couple of decades.) This is the essence of theplanar rooted tree operad [46].

·· ··1 2 i k

�i

· · ·1 2 l

=

· · ·· · · ··· · · · · · · · ·1 i j n

Figure 3: The grafting Mk �i Ml of the l –corolla Ml to the i -th leaf ofk –corolla Mk , where j D i C l � 1 and nD kC l � 1

Multilinear maps compose in just this way, so relations (5) can be phrased as saying wehave a map from planar rooted trees to multilinear maps respecting the �i ‘products’,the essence of a map of operads [46]. This was originally observed in terms of thevector spaces of chains on a based loop space, but abstracted as follows: Let A1.n/,n � 1, be a graded vector space spanned by planar rooted trees with n leaves withidentity e 2 A1.1/. For a planar rooted tree T 2 A1.n/, its grading is introducedas the number of the vertices contained in T , which we denote by v.T /. A treeT 2A1.n/, n� 2, with v.T /D 1 is the corolla Mn . Any tree T with v.T /D 2 isobtained by the grafting of two corollas as in Figure 3. Grafting of any two trees isdefined in a similar way, with an appropriate sign rule, and any tree T with v.T /� 2

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Homotopy algebra of open–closed strings 235

can be obtained recursively by grafting a corolla to a tree T 0 with v.T 0/D v.T /� 1.One can define a differential d of degree one, which acts on each corolla as

(3) d .Mn/D�X

k;l�2; kClDnC1

kXiD1

Mk �i Ml

and extends to one on A1 WD ˚n�1A1.n/ by the following rule:

d.T �i T 0/D d.T / �i T 0C .�1/v.T /T �i d.T 0/ :

If we introduce the contraction of internal edges, that is, indicate by T 0! T that T isobtained from T 0 by contracting an internal edge, the differential is equivalently givenby

d.T /DX

T 0!T

˙T 0

with an appropriate sign ˙. Thus, one obtains a dg operad A1 , which is known asthe A1–operad.

An algebra A over A1 is obtained by a representation �W A1.k/! Hom.A˝k ;A/,ie, a map � compatible with the �i ’s and also the differentials. We denote by mk theimage �.Mk/ of Mk by � . Then, for each corolla we have

(4)X

kClDnC1

kXiD1

˙mk �i ml D 0;

where we now write m1 for d .

This then becomes the definition.

Definition 2.1 (A1–algebra – strongly homotopy associative algebra – [53]) Let A

be a Z–graded vector space AD˚r2ZAr and suppose that there exists a collection ofdegree one multilinear maps

m WD fmk W A˝k!Agk�1 :

.A;m/ is called an A1–algebra when the multilinear maps mk satisfy the followingrelations

(5)X

kClDnC1

kXiD1

˙mk.o1; : : : ; oi�1;ml.oi ; : : : ; oiCl�1/; oiCl ; : : : ; on/D 0

for n� 1.

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236 Hiroshige Kajiura and Jim Stasheff

A weak or curved A1–algebra consists of a collection of degree one multilinear maps

m WD fmk W A˝k!Agk�0

satisfying the above relations, but for n� 0 and in particular with k; l � 0.

Remark 2.2 The ‘weak’ version is fairly new, apparently first in papers of Getzlerand Jones (and Petrack) [18; 19], then was adopted by Zwiebach in the L1–context[61] and later used to study what physicists refer to as a ‘background’ for string fieldtheory. The map m0W C!A is regarded as an element m0.1/ 2A. The augmentedrelation then implies that m0.1/ is a cycle, but m1m1 need no longer be 0, ratherm1m1 D˙m2.m0˝ 1/˙m2.1˝m0/.

Remark 2.3 Recall, as mentioned earlier, that the component maps would have mk

of degree .k � 2/ in the original formulation.

Definition 2.4 (A1–morphism) For two A1–algebras .A;m/ and .A0;m0/, a col-lection of degree zero (degree preserving) multilinear maps

ffk W A˝k!A0gk�1

is called an A1–morphism ffkgk�1W .A;m/! .A0;m0/ iff it satisfies the followingrelations: X

1�k1<k2���<kjDn

m0j .fk1.o1; ::; ok1

/; fk2�k1.ok1C1; ::; ok2

/;

: : : ; fn�kj�1.okj�1C1; ::; on//

D

XkClDnC1

kXiD1

˙fk.o1; : : : ; oi�1;ml.oi ; ::; oiCl�1/; oiCl ; : : : ; on/

(6)

for n � 1. In particular, if f1W A! A0 induces an isomorphism between the coho-mologies H.A/ and H.A0/, the A1–morphism is called an A1–quasi-isomorphism.

A1–quasi-isomorphisms play important roles from the homotopy algebraic point ofview (see Section 3).

2.4 Coalgebra formulation

The maps mk can be assembled into a single map, also denoted m, from the tensorspace T cAD˚k�0A˝k to A with the convention that A˝0DC. The grading implied

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Homotopy algebra of open–closed strings 237

by having the maps mk all of degree one is the usual grading on each A˝k : We canregard T cA as the tensor coalgebra by defining

4.o1˝ � � �˝ on/D†npD0.o1˝ � � �˝ op/˝ .opC1˝ � � �˝ on/ :

A map f 2Hom.T cA;T cA/ is a graded coderivation means 4f D .f ˝1C1˝f /4,with the appropriate signs and dual to the definition of a graded derivation of analgebra. Here 1 denotes the identity 1W A!A. We then identify Hom.T cA;A/ withCoder.T cA/ by lifting a multilinear map as a coderivation [55]. Analogously to thesituation for derivations, the composition graded commutator of coderivations is againa coderivation; this graded commutator corresponds to the Gerstenhaber bracket onHom.T cA;A/ [14; 55]. Notice that this involves a shift in grading since Gerstenhaberuses the traditional Hochschild complex grading. Thus Coder.T cA/ is a graded Liealgebra and in fact a dg Lie algebra with respect to the bar construction differential,which corresponds to the Hochschild differential on Hom.T cA;A/ in the case of anassociative algebra .A;m/ [14]. Using the bracket, the differential can be written asŒm; �.

The advantage of this point of view is that the component maps mk assemble into asingle map m in Coder.T cA/ and relations (5) can be summarized by

Œm;m�D 0 or; equivalently; D2D 0 ;

where D D Œm; �. In fact, m 2 Coder.T cA/ is an A1–algebra structure on A iffŒm;m�D 0 and m has no constant term, m0 D 0. If m0 ¤ 0, the structure is a weakA1–algebra. The A1–morphism components similarly combine to give a singlemap of dg coalgebras fW T cA! T cA0 , .f˝ f/4D4f. In particular, equation (6) isequivalent to f ımDm0 ı f.

2.5 L1–algebras

Since an ordinary Lie algebra g is regarded as ungraded, the defining bracket is regardedas skew-symmetric. If we regard g as all of degree one, then the bracket would begraded symmetric. For dg Lie algebras and L1–algebras, we need graded symmetry,which refers to the usual symmetry with signs determined by the grading. The basicrelation is

(7) � W x˝y 7! .�1/jxjjyjy˝x :

The sign of a permutation of n graded elements, is defined by

(8) �.c1; : : : : ; cn/D˙.c�.1/; : : : : ; c�.n//;

where the sign ˙ is given by what is called the Koszul sign of the permutation.

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238 Hiroshige Kajiura and Jim Stasheff

Definition 2.5 (Graded symmetry) A graded symmetric multilinear map of a gradedvector space V to itself is a linear map f W V ˝n ! V such that for any ci 2 V ,1� i � n, and any � 2Sn (the permutation group of n elements), the relation

f .c1; : : : : ; cn/D˙f .c�.1/; : : : : ; c�.n//

holds, where ˙ is the Koszul sign above.

The graded symmetric coalgebra C.V / on a graded vector space V is defined as thesubcoalgebra C.V /�T cV consisting of the graded symmetric elements in each V ˝n .

Definition 2.6 By a .k; l/–unshuffle of c1; : : : ; cn with nD kC l is meant a permuta-tion � such that for i < j �k , we have �.i/<�.j / and similarly for k< i < j �kCl .We denote the subset of .k; l/–unshuffles in SkCl by Sk;l and by SkClDn; the unionof the subgroups Sk;l with kC l D n.

Definition 2.7 (L1–algebra (strong homotopy Lie algebra) [44]) Let L be a gradedvector space and suppose that a collection of degree one graded symmetric linear mapsl WD flk W L

˝k !Lgk�1 is given. .L; l/ is called an L1–algebra iff the maps satisfythe following relations

(9)X

�2SkClDn

˙l1Cl.lk.c�.1/; : : : ; c�.k//; c�.kC1/; : : : ; c�.n//D 0

for n� 1, where ˙ is the Koszul sign (8) of the permutation � 2SkClDn .

A weak L1–algebra consists of a collection of degree one graded symmetric linearmaps l WD flk W L

˝k !Lgl�0 satisfying the above relations, but for n � 0 and withk; l � 0.

Remark 2.8 The alternate definition in which the summation is over all permutations,rather than just unshuffles, requires the inclusion of appropriate coefficients involvingfactorials.

Remark 2.9 A dg Lie algebra is expressed as the desuspension of an L1–algebra.L; l/ where l1 and l2 correspond to the differential and the Lie bracket, respectively,and higher multilinear maps l3; l4; : : : are absent.

Remark 2.10 For the ‘weak/curved’ version, remarks analogous to those for weakA1–algebras apply, and similarly for morphisms.

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Homotopy algebra of open–closed strings 239

In a similar way as in the A1 case, an L1–algebra .L; l/ is described as a coderivationlW C.L/! C.L/ satisfying .l/2 D 0. Also, for two L1–algebras .L; l/ and .L0; l0/,an L1–morphism is defined as a coalgebra map fW C.L/! C.L0/, where f consistsof graded symmetric multilinear maps fk W L

˝k ! L0 of degree zero with k � 1,satisfying l0 ı fD f ı l.

The tree operad description of L1–algebras uses non-planar rooted trees with leavesnumbered 1; 2; : : : arbitrarily [46]. Namely, a non-planar rooted tree can be expressedas a planar rooted tree but with arbitrary ordered labels for the leaves. In particular,corollas obtained by permuting the labels are identified (Figure 4). Let L1.n/, n� 1,

· · ·1 2 3 k

D

· · ·�.1/ �.2/ �.3/ �.k/

Figure 4: Nonplanar k –corolla Lk corresponding to lk . Since edges arenon-planar, it is symmetric with respect to the permutation of the edges.

be a graded vector space generated by those non-planar rooted trees of n leaves. For atree T 2L1.n/, a permutation � 2Sn of the labels for leaves generates a different treein general, but sometimes the same one because of the symmetry of the corollas above.The grafting, ıi , to the leaf labelled i is defined as in the planar case in subsection 2.3,and any non-planar rooted tree is obtained by grafting corollas fLkgk�2 recursively,as in the planar case, together with the permutations of the labels for the leaves. Adegree one differential d W L1.n/! L1.n/ is given in a similar way; for T 0! T

indicating that T is obtained from T 0 by the contraction of an internal edge,

d.T /DX

T 0!T

˙T 0 ;

and d.T ıi T 0/ D d.T / ıi T 0 C .�1/v.T /T ıi d.T 0/ again holds. Thus, L1 WD˚n�1L1.n/ forms a dg operad, called the L1–operad.

An algebra L over L1 obtained by a map �W L1.k/! Hom.L˝k ;L/ then formsan L1–algebra .L; l/.

2.6 Compactification of moduli spaces of spheres with punctures

As A1–algebras can be described in terms of compactifications of moduli spaces ofconfigurations of points on an interval, so, with some additional subtlety, L1–algebras

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240 Hiroshige Kajiura and Jim Stasheff

can be described in terms of compactifications of moduli spaces of configurations ofpoints on a Riemann sphere. The compactification corresponding to an L1–structureis the real compactification M0;n of the moduli spaces M0;n of spheres with n

punctures ([40], see also [61]). Here we use underbar in order to distinguish it from thecomplex compactification by Deligne–Knudsen–Mumford which is more familiar andoften denoted by xM0;n . Also, we attach the lower index 0 indicating genus zero, inorder to distinguish the real compactification of the moduli spaces of punctured spheresfrom that of punctured disks in subsection 2.2.

The moduli space M0;n is defined as the configuration space of n points on a sphere' C[f1g modulo the SL.2;C/ action

w0.w/DawC b

cwC d; w 2 C[f1g ;

�a b

c d

�2 SL.2;C/:

This SL.2;C/ action allows us to fix three points; usually 0; 1 and 1.

For n D 3, the moduli space M0;2C1 is a point, so is its real compactificationM0;2C1 ' fptg. For nD 4, one has

M0;4 ' .C[1/�f0; 1;1g;

which is the configuration space of four points 0; 1; w;1 with the subtraction ofthe ‘diagonal’. The real compactification of M0;4 looks as in Figure 5: M0;4 has

0 1

1B0 B1

B1

'

B0 B1

B1

Figure 5: The real compactification M0;3C1 of M0;3C1

codimRD 1 boundaries B0 , B1 , B1 . If we associate points 0; 1; w to x;y; z and 1

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Homotopy algebra of open–closed strings 241

to the root edge, we get the correspondence:

B0 $ ˙ŒŒx; z�;y�

B1 $ ˙ŒŒy; z�;x�

B1 $ ˙ŒŒx;y�; z� :

Inspired by closed string field theory, this can be seen in terms of ‘grafting’ tubularneighborhoods of trees with freedom of a full S1 of rotations of the boundaries whichare to be identified:

x y

z

1

$

x y

z

1

S1

Now consider the relative homology groups of the compactified moduli spaces mod-ulo those on the lower dimensional strata. These give a version of the L1–operad.Corresponding to the relation @.M0;4/D B0

`B1

`B1 , we obtain:

d.l3/.x;y; z/D ŒŒx;y�; z�˙ ŒŒy; z�;x�˙ ŒŒz;x�;y� :

Notice that M0;n is not contractible for n� 4. In general, M0;n is a manifold withcorners (as were the associahedra) of real dimension 2n� 6, but the strata are notgeneraly cells, as they were for the associahedra. Thus, to define the dg L1–operad,we use the homology of strata relative to boundary [40]. On the other hand, if we areconcerned only with the corresponding homology operad, we need only use the littledisks operad and Fred’s configuration space calculations [7].

2.7 Open–closed homotopy algebra (OCHA)

For our open–closed homotopy algebra, we consider a graded vector space HDHc˚Ho

in which Hc will be an L1–algebra and Ho , an A1–algebra.

An OCHA is inspired by the compactification of the moduli spaces of puncturedRiemann surfaces (Riemann surfaces with marked points) or the decomposition of the

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242 Hiroshige Kajiura and Jim Stasheff

moduli spaces as is done in constructing string field theory. More precisely, an OCHAshould be an algebra over the DG operad of chains of the compactified moduli spacesof the corresponding Riemann surfaces. In this paper, we first present the DG–operadwhich we call the open–closed operad OC1 . An OCHA obtained as a representationof OC1 has various interesting structures also from purely algebraic points of view.In particular, an OCHA can be viewed as a generalization of various known algebras.We shall discuss this after presenting the definition of an OCHA. Before giving theexplicit definition in terms of ‘algebraic’ formulas, we look at the tree description.

2.8 The tree description

We associated the k –corolla Mk of planar rooted trees to the multilinear map mk of anA1–algebra, and the k –corolla Lk of non-planar rooted trees to the graded symmetricmultilinear map lk of an L1–algebra. For an OCHA .H; l; n/, we introduce the.k; l/–corolla Nk;l

(10) Nk;l D

· · · · · ·· · ·· · ·1 1k l

which is defined to be partially symmetric (non-planar); only symmetric with respectto the k leaves. We express symmetric leaves as wiggly edges and planar (= non-symmeteric) leaves as straight edges as before. Let us consider such corollas for2kC l C 1 � 3. As we shall explain further later, this constraint is motivated by thestable moduli space of a disk with k –punctures interior and .l C 1/–punctures onthe boundary of the disk. We also consider non-planar corollas fLkgk�2 . The planark –corolla Mk is already included as N0;k . Since we have two kinds of edges, wehave two kinds of grafting. We denote by ıi (resp. �i ) the grafting of a wiggly (resp.straight) root edge to an i -th wiggly (resp. straight) edge, respectively. For thesecorollas, we have three types of composite; in addition to the composite L1Ck ıi Ll inL1 , there is a composite Nk;m ıi Lp described by

· · · · · ·· · ·· · ·1 1k m

ıi

· · ·1 2 3 p

D

· · · · · · · · ·[ ] [ ] ( )· · · · · · · · ·

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Homotopy algebra of open–closed strings 243

where in the right hand side the labels are given by Œi; : : : ; i Cp� 1�Œ1; : : : ; i � 1; i C

p; : : : ;pC k � 1�.1; : : : ;m/, and the composite Np;q �i Nr;s

· · · · · ·· · ·· · ·1 1p q

�i

· · · · · ·· · ·· · ·1 1r s

D

···· ·· ····[ ] ( ) [ ] ( ) ( )· · · · · · · · · · · · · · ·

with labels Œ1; : : : ;p�.1; : : : ; i�1/ŒpC1; : : : ;pCr �.i; : : : ; iCs�1/.iCs; : : : ; qCs�1/.To these resulting trees, grafting of a corolla Lk or Nk;l can be defined in a naturalway, and we can repeat this procedure. Let us consider tree graphs obtained in thisway, that is, by grafting the corollas lk and nk;l recursively, together with the actionof permutations of the labels for closed string leaves. Each of them has a wiggly orstraight root edge. The tree graphs with wiggly root edges, with the addition of theidentity ec 2 L1.1/, generate L1 as stated in subsection 2.5. On the other hand, thetree graphs with both wiggly and straight edges are new.

Definition 2.11 We denote by N1.kI l/, the graded vector space generated by rootedtree graphs with k wiggly leaves and l straight leaves. In particular, we formally addthe identity eo generating N1.0I 1/ and a corolla N1;0 generating N1.1I 0/. Thetree operad relevant for OCHAs is then OC1 WD L1˚N1 .

In fact, OC1 is an example of a colored operad [5; 46; 59]. For each tree T 2OC1 ,its grading is given by the number of vertices v.T /.

For trees in OC1 , let T 0! T indicate that T is obtained from T 0 by contracting awiggly or a straight internal edge. A degree one differential d W OC1!OC1 is givenby

(11) d.T /DX

T 0!T

˙T 0 ;

so that the following compatibility holds:

d.T ıi T 0/D d.T / ıi T 0C .�1/v.T /T ıi d.T 0/;

d.T �i T 00/D d.T / �i T 00C .�1/v.T /T �i d.T 00/ :

Thus, OC1 forms a dg colored operad.

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2.9 Formal definition of OCHA

For two Z–graded vector spaces Hc and Ho , an open–closed homotopy algebra.H WDHc˚Ho; l; n/ is an algebra over the operad OC1 . An algebra H WDHc˚Ho

over OC1 is obtained by a representation

�W L1.k/! Hom.H˝kc ;Hc/ ; �W N1.kI l/! Hom..Hc/

˝k˝ .Ho/

˝l ;Ho/

which is compatible with respect to the grafting ıi , �i and the differential d . Here,regarding elements in both Hom.H˝k

c ;Hc/ and Hom..Hc/˝k˝.Ho/

˝l ;Ho/ as thosein Coder.C.Hc/˝T c.Ho//, the differential on the algebra side is given by

(12) d WD Œl1C n0;1; �;

where both l1 and n0;1 are the canonical lift of differentials l1W Hc ! Hc andn0;1W Ho ! Ho on the corresponding graded vector spaces. On the other hand, inaddition to the differential d W L1 ! L1 defining the L1–structure, we have thedifferential d W N1!N1 (11) which acts on the corolla Nk;l as

d.Nn;m/DX

kCpDnC1

Xi

Nk;m ıi LpC

XpCrDn;qCsDmC1

Xi

Np;q �i Nr;s:

By combining this with equation (12), one can write down the conditions for an OCHA:

0DX

�2SpCrDn

˙n1Cr;m

�.lp˝ 1˝r

c ˝ 1˝mo /.c�.I /I o1; : : : ; om/

�C

X�2SpCrDn

iCsCjDm

˙np;iC1Cj

�.1˝p

c ˝ 1˝io ˝ nr;s˝ 1˝j

o /.c�.I /I o1; : : : ; om/�;

where c1; : : : ; cn and o1; : : : ; om are homogeneous elements in Hc and Ho , respec-tively, and the signs ˙ are the Koszul sign (8) of � .

More explicitly:

Definition 2.12 (Open–Closed Homotopy Algebra (OCHA) [37]) An open–closedhomotopy algebra (OCHA) .HDHc˚Ho; l; n/ consists of an L1–algebra .Hc ; l/

and a family of maps nD fnp;qW H˝pc ˝H˝q

o !Hog of degree one for p; q � 0 with

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Homotopy algebra of open–closed strings 245

the exception of .p; q/D .0; 0/ satisfying the compatibility conditions:

0DX

�2SpCrDn

˙n1Cr;m.lp.c�.1/; : : : ; c�.p//; c�.pC1/; : : : ; c�.n/I o1; : : : ; om/

+X

�2SpCrDn; iCsCjDm

˙np;iC1Cj .c�.1/; ::; c�.p/I o1; ::; oi ;

nr;s.c�.pC1/; ::; c�.n/I oiC1; ::; oiCs/; oiCsC1; ::; om/

(13)

for homogeneous elements c1; : : : ; cn 2Hc and o1; : : : ; om 2Ho with the full rangen;m� 0, .n;m/¤ .0; 0/. The signs ˙ are given in [37].

A weak/curved OCHA consists of a weak L1–algebra .Hc ; l/ with a family of mapsnD fnp;qW H˝p

c ˝H˝qo !Hog, of degree one, now for p; q � 0 satisfying the analog

of the above relation.

An open–closed homotopy algebra includes various sub-structures or reduces to varioussimpler structures as particular cases. The substructure .Hc ; l/ is by definition anL1–algebra and .Ho; fmk WD n0;kg/ forms an A1–algebra. A nontrivial structureobtained as a special case of OCHAs is the action of Hc as an L1–algebra on Ho asa dg vector space. Lada and Markl ([43] Definition 5.1) provide the definition of anL1–module where one can see the structure as satisfying the relations for a Lie module‘up to homotopy’. This is the appropriate strong homotopy version of the action of anordinary Lie algebra L on a vector space M , also described as M being a moduleover L or a representation of L. If we set np;0 D 0 for all p � 1, the substructure.H; fnp;1g/ makes Ho an L1–module over .Hc ; l/. Thus we can also speak of Ho

as a strong homotopy module over Hc or as a strong homotopy representation of Hc

(cf [54]).

On the other hand, n1;q with q � 1 forms a strong homotopy derivation [45] withrespect to the A1–algebra .Ho; fmkgk�1/. Moreover, we have the strong homotopyversion of an algebra A over a Lie algebra L, that is, an action of L by derivations ofA, so that the L1–map L! End.A/ takes values in the Lie sub-algebra DerA.

In his ground breaking “Notions d’algebre differentielle; � � � ” [6], Henri Cartanformalized several dg algebra notions related to his study of the deRham cohomologyof principal fibre bundles, in particular, that of a Lie group G acting in (‘dans’) adifferential graded algebra E. The action uses only the Lie algebra g of G . Cartan’saction includes both the graded derivation d , the Lie derivative �.X / and the innerderivative i.X / for X 2 g. The concept was later reintroduced by Flato, Gerstenhaberand Voronov [11] under the name Leibniz pair.

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246 Hiroshige Kajiura and Jim Stasheff

We need only the �.X / (which we denote �.X / since by � we denote the image by �of an element X 2 g), then the algebraic structure is an example of a dg algebra over adg Lie algebra g. Its higher homotopy extension a mathematician would construct bythe usual procedures of strong homotopy algebra leads to the following definition (seethe Appendix by M Markl in [37]):

Definition 2.13 (A1–algebra over an L1–algebra) Let L be an L1–algebra andA an A1–algebra which as a dg vector space is an sh-L module. That A is an A1–algebra over L means that the module structure map �W L! End.A/, regarded as inCoder.T cA/, extends to an L1–map L! Coder.T cA/.

An A1–algebra over an L1–algebra defined as above is an OCHA .H; l; n/ withnp;0 D 0 for p � 1.

Given two OCHAs .H; l; n/ and .H0; l0; n0/, an OCHA morphism from .H; l; n/ to.H0; l0; n0/ is defined by a collection of degree zero multilinear maps fk W .Hc/

˝k!H0c ,k � 1, and fk;l W .Hc/

˝k˝ .Ho/˝l !H0o , k; l > 0, .k; l/¤ .0; 0/, satisfying certain

conditions [37]. In particular, ffkgk�1 forms an L1–morphism from .Hc ; l/ to.H0c ; l0/. The notion of OCHA–quasi-isomorphisms is defined as OCHA–morphismssuch that both f1W Hc!H0c and f0;1W Ho!H0o induce isomorphisms on the coho-mologies.

An OCHA .H; l; n/ has a coalgebra description in terms of a degree one codifferentialconstructed from l and n on the tensor coalgebra of H (see [37]). Hoefel [30] hasrecently shown the following:

Theorem 2.14 (Hoefel [30]) OCHAs are characterized as being given by all coderiva-tions of degree 1 and square zero on Coder.C.Hc/˝T c.Ho//.

Then, two OCHAs .H; l; n/ and .H0; l0; n0/, an OCHA–morphism fW .H; l; n/ !.H0; l0; n0/ is described as a coalgebra map fW C.Hc/˝T c.Ho/! C.H0c/˝T c.H0o/satisfying .l0C n0/ ı fD f ı .lC n/, where lC n is the codifferential constructed fromthe OCHA structures l and n.

Also, one can describe an OCHA dually in terms of a supermanifold, see [38].

2.10 Examples of the moduli space description

An OCHA should be an algebra over the DG operad of relative chains of the compactifiedmoduli spaces of the corresponding punctured Riemann surfaces. The strata of the

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Homotopy algebra of open–closed strings 247

compactified moduli space can be labelled by the trees of the OC1–operad. In thisdirection, Hoefel discusses more carefully the details of these structures [29].

Let us consider a moduli space corresponding to the open–closed case. For np;q , thecorresponding moduli space is that of a disk with p–punctures in the bulk (interior)and .qC1/–punctures on the boundary. For pD 0, as we saw, fmq D n0;qg forms anA1–structure, and the corresponding moduli spaces are the associahedra. The modulispace corresponding to the operation fn1;qg with one closed string input is the same asthe cyclohedra fWqC1g, which is the moduli space of configuration space of points onS1 modulo rotation discussed by Bott and Taubes (see [46, page 241] and [57]).

The cyclohedra fWng are contractible polytopes. However, the moduli spaces cor-responding to np;q with p � 2 are not contractible in general. Let us consider themoduli space corresponding to n2;q : the disk with two two interior punctures (= closedstrings). For q D 0, the moduli space is described as in Figure 6 (a), which is whatis called ‘The Eye’ in the paper on deformation quantization by Kontsevich [42]. Forn2;1 , the compactified moduli space is topologically a solid torus as in Figure 6 (b)(this figure was made by S Devadoss [9]).

(a) (b)

W2

W2

S1

Figure 6: (a) The Eye (b) The solid torus

Recall that the facets (codim one faces) of the associahedra are products of associahedra.For cyclohedra, the facets are products of cyclohedra and associahedra. The analogholds for the open–closed compactified moduli spaces (which currently are nameless),although they are not polytopes.

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2.11 Cyclic structure

We can also consider an additional cyclic structure on open–closed homotopy algebras.See [37; 38] for the definition. The cyclic structure can be defined in terms of symplecticinner products. These inner products are essential to the description of the Lagrangiansappearing in string field theory (see [36]). The string theory motivation for this additionalstructure is that punctures on the boundary of the disk inherit a cyclic order from theorientation of the disk and the operations are to respect this cyclic structure, just as theL1–structure reflects the symmetry of the punctures in the interior of the disk or onthe sphere.

Let us explain briefly the cyclicity in the case of A1–algebras. In terms of trees, thedistinction between the root and the leaves can be absorbed by regarding these edgesas cyclically ordered.

From the viewpoint of the moduli spaces MnC1 of punctured disks, representingMnC1 as in equation (2), the cyclic action is an automorphism gWMnC1!MnC1 ,where g is an SL.2;R/ transformation (1) such that

.1; 0D t1; t2; : : : ; tn�1; tn D 1/ 7!.g.1/;g.0/;g.t2/; : : : ;g.tn�1/;g.1//

D .0; t2; : : : ; tn�1; 1;1/:

One can compactify MnC1 so that the cyclic action extends to the one on xMnC1 . Inthis way, one can consider a cyclic action on the associahedra. This cyclic action forthe associahedra is discussed in [20]. Thus, from the viewpoint of Riemann surfaces, ie,string theory, taking the cyclic action into account for the associahedra is very natural.

This can also be seen visually from the planar trees associated to disks with points onthe boundary, cf Figure 2 (b). Correspondingly, for an A1–algebra .A;m/, a cyclicstructure is defined by a (nondegenerate) inner product !W A˝A! C of fixed integerdegree satisfying

!.mn.o1; : : : ; on/; onC1/D˙!.mn.o2; : : : ; on; onC1/; o1/

for any homogeneous elements o1; : : : ; onC1 2A (see [46]).

In a similar way using a nondegenerate inner product, a cyclic structure is defined foran L1–algebra and then for an OCHA [37; 38].

3 The minimal model theorem

Homotopy algebras are designed to have homotopy invariant properties. A key anduseful theorem in homotopy algebras is then the minimal model theorem, proved by

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Homotopy algebra of open–closed strings 249

Kadeishvili for A1–algebras [34]. For an A1–algebra .A;m/, the minimal modeltheorem states that there exists another A1–algebra .H.A/;m0/ on the cohomologiesof .A;m1/ and an A1–quasi-isomorphism from .H.A/;m0/ to .A;m/. Since m0 isan A1–structure on the cohomology H.A/, the differential m0

1is trivial; such an

A1–algebra is called minimal.

The minimal model theorem holds also for an OCHA, which implies that an OCHA isalso appropriately called a homotopy algebra:

Definition 3.1 (Minimal open–closed homotopy algebra) An OCHA .H D Hc ˚

Ho; l; n/ is called minimal if l1 D 0 on Hc and n0;1 D 0 on Ho .

Theorem 3.2 (Minimal model theorem for open–closed homotopy algebras) For agiven OCHA .H; l; n/, there exists a minimal OCHA .H.H/; l0; n0/ and an OCHA–quasi-isomorphism fW .H.H/; l0; n0/! .H; l; n/.

(See subsection 2.9 for the definition of an OCHA–quasi-isomorphism. )

Various stronger versions of this minimal model theorem hold for OCHAs [37], asfor A1–algebras, L1–algebras, etc. One of them is the homological perturbationtheory developed in particular on the homology of a differential graded algebra [23;32; 27; 24; 25; 26] (see [47] for an application) and of a dg Lie algebra [33]. Anotherone is the decomposition theorem (see [39; 36]). As for classical A1 , L1 , etc cases,these theorems imply that OCHA–quasi-isomorphisms and in particular the one inTheorem 3.2 in fact give homotopy equivalence between OCHAs. This further impliesthe uniqueness of a minimal model for an OCHA .H; l; n/; a minimal OCHA H.H/is unique up to an isomorphism on H.H/.

4 Geometric construction of OCHAs and Merkulov’s struc-tures

As mentioned, OCHAs admit a geometric expression in terms of supermanifolds as givenin [38]. Here, instead of that, we give a partially supermanifold description in which anOCHA can be viewed as a ‘geometric’ weak A1–structure. This description is inspiredby Merkulov’s geometric A1– (and C1 )–structures discussed as a generalization ofFrobenius structures [48].

Definition 4.1 (Merkulov [49; 48] – paraphrased) A (Merkulov) geometric A1–structure on a graded manifold M with its tangent bundle TM is a collection of mapsfor n� 1:

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250 Hiroshige Kajiura and Jim Stasheff

(i) �nW ˝nOM

TM ! TM ,

(ii) where �1 WD Œ�; � is defined in terms of an element � 2 TM such that Œ�; ��D 0,making the sheaf of sections TM of TM into a sheaf of A1–algebras.

Condition (ii) means that � is a homological vector field on TM , cf the dual superman-ifold description of an A1–structure.

The Merkulov geometric A1–structure can be obtained as a special case of an OCHA.H WDHc ˚Ho; l; n/ in which the Z–graded supermanifold is an L1–algebra Hc .With an eye toward the relevant deformation theory, we use the formal graded commuta-tive power series ring denoted by CŒŒ ��. More precisely, denote by feig a basis of Hc

and the dual base as i , where the degree of the dual basis is set by deg. i/D�deg.ei/.Then CŒŒ �� is the formal graded power series ring in the variables f ig.

Let us express the L1–structure lk W .Hc/˝k !Hc , in terms of the bases:

lk.ei1; : : : ; eik

/D ej cji1���ik

:

Correspondingly, let us define an odd formal vector field on Hc , that is, a derivation ofCŒŒ ��:

(14) ıS D

�@

@ jcj . /D

Xk�0

1

k!

�@

@ jc

ji1���ik

ik � � � i1 ; cj . / 2 CŒŒ �� :

Also, define a collection of multilinear structures on Ho parameterized by Hc asfollows:

(15) ni1;:::;ipIq.o1; : : : ; oq/ WD np;q.ei1; : : : ; eip I o1; : : : ; oq/

for p; q � 0 with pC q > 0. Then, let us define a new collection of multilinear mapson zHo WDHo˝CŒŒ �� as follows:

zmn WD

Xk�0

ni1���ip;q ip � � � i1 W . zHo/

˝n! zHo ; .n¤ 1/ ;(16)

zm1.zo/ WDXk�0

ni1���ip;q ip � � � i1.zo/� ıS .zo/;(17)

where the tensor product zo˝n is defined over CŒŒ ��. One can see that . zHo; f zmkgk�0/

forms a weak A1–algebra over CŒŒ ��. In fact, generalized to a more general basemanifold M, it may be plausible to call this a geometric weak A1–structure moregeneral than Merkulov’s, which can be regarded as the case in which Ho is the fiberof T0Hc , the tangent space of Hc at the origin of Hc and then we drop the secondcondition of his geometric A1–structure. Clearly an isomorphism of T0Hc to Hc

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Homotopy algebra of open–closed strings 251

as graded vector spaces extends to an isomorphism from THcto zHc WDHc˝CŒŒ ��,

cf [49], subsection 3.8.1. Under this identification, let us consider the particular casenp;1 WD

1.pC1/!

lpC1 . Thus, the differential zm1 in equation (17) turns out to be

zm1.zc/D�ŒıS ; zc�; zc 2 zHc :

Since he does not treat the ‘weak’ case, np;0 is of course zero for any p . The highermultilinear maps zmnW . zHc/

˝n! zHc , n� 2, are defined in the same way as in equation(16), only with the replacement of elements in zHo by those in zHc . One can see that. zHc ; zm/ obtained as above coincides with the geometric A1–structure in subsection3.8.1 of [49]. Then, the theorem in subsection 3.8.2 in [49] states that a geometricA1–structure is equivalent to a Gerst1–algebra structure which is defined by certainrelations described by tree graphs having both straight edges and wiggly edges as insubsection 3.6.1 of [49]. Unfortunately, there a straight (resp. wiggly) edge correspondsto an element in zHo (resp. zHc ), so his convention is the opposite of ours for OCHAs.Even taking this into account, his defining equation for a Gerst1–algebra structure issuperficially different from ours. This is because we have np;1 D

1.pC1/!

lpC1 now; inthe Gerst1–algebra case, the L1–structure lk (which is denoted by �k in [49]) isour lk or our k!nk�1;1 . Remembering this fact, one can see that the Gerst1–algebracondition in [49] is a special case of our OCHA condition.

The commutative version of the geometric A1–structure is called a geometric C1–structure [49; 48], which is a special G1–algebra and plays an important role indeformation theory. For a given geometric C1–structure, if we concentrate on thedegree zero part of the graded vector space Hc , the higher products are also concentratedon the one with degree zero. The resulting special geometric C1–structure is what iscalled an F –manifold, a generalization of a Frobenius manifold.

5 Applications of OCHAs to deformation theory

Consider an OCHA .HDHc ˚Ho; l; n/. We will show how the combined structureimplies the L1–algebra .Hc ; l/ controls some deformations of the A1–algebra.Ho; fmkgk�1/. We will further investigate the deformations of this control as H isdeformed.

We first review some of the basics of deformation theory from a homotopy algebra pointof view. The philosophy of deformation theory which we follow (due originally, webelieve, to Grothendieck 1 cf [52; 21; 8]) regards any deformation theory as ‘controlled’by a dg Lie algebra g (unique up to homotopy type as an L1–algebra).

1See [10] for an extensive annotated bibliography of deformation theory.

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252 Hiroshige Kajiura and Jim Stasheff

For the deformation theory of an (ungraded) associative algebra .A;m/ [15] the standardcontrolling dg Lie algebra is Coder.T cA/ with the graded commutator as the graded Liebracket [55]. Under the identification (including a shift in grading) of Coder.T cA/ withHom.T cA;A/ (which is the Hochschild cochain complex), this bracket is identifiedwith the Gerstenhaber bracket and the differential with the Hochschild differential,which can be written as Œm; � [14].

The generalization to a differential graded associative algebra is straightforward; thedifferential is now: ŒdA Cm2; �. For an A1–algebra, the differential similarlygeneralizes to Œm; �.

Deformations of A correspond to certain elements of Coder.T cA/, namely those thatare solutions of an integrability equation, now known more commonly as a Maurer–Cartan equation.

Definition 5.1 (The classical Maurer–Cartan equation) In a dg Lie algebra .g; d; Œ ; �/,the classical Maurer–Cartan equation is

(18) d� C1

2Œ�; � �D 0 for � 2 g1:

For an A1–algebra .A;m/ and � 2Coder1.T cA/, a deformed A1–structure is givenby mC � iff

.mC �/2 D 0 :

Teasing this apart, since we start with m2 D 0, we have equivalently

(19) D� C1

2Œ�; � �D 0;

the Maurer–Cartan equation of the dg Lie algebra .Coder.T cA/;D; Œ ; �/ (Here D isthe natural differential on Coder.T cA/� End.T cA/, ie D� D Œm; � �. )

For L1–algebras, the analogous remarks hold, substituting the Chevalley–Eilenbergcomplex for that of Hochschild, ie, using Coder C.L/' Hom.C.L/;L/.

Definition 5.2 (The strong homotopy Maurer–Cartan equation) In an L1–algebra.L; l/, the (generalized) Maurer–Cartan equation isX

k�1

1

k!lk.xc; : : : ; xc/D 0

for xc 2L0 . 2

2Note that the degree of xc is zero since a dg Lie algebra is precisely a special L1–algebra after asuitable degree shifing called the suspension and then g1 DL0 .

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Homotopy algebra of open–closed strings 253

We denote the set of solutions of the Maurer–Cartan equation as MC.L; l/ or moresimply MC.L/.Now, since an OCHA can be thought of as a generalization of an A1–algebra over anL1–algebra (Definition 2.13), one has:

Theorem 5.3 [37; 38] An OCHA .H WD Hc ˚Ho; l; n/ is equivalent to an L1–morphism from .Hc ; l/ to .Coder.T c.Ho//;D D Œm; �; Œ ; �/, where m is the codifer-ential on Coder.T c.Ho// corresponding to fmk D n0;kgk�1 .

Since it is known that an L1–morphism preserves the solutions of the Maurer–Cartanequations, we obtain the following:

Theorem 5.4 For an OCHA .H WD Hc ˚Ho; l; n/, a Maurer–Cartan element xc 2M.Hc ; l/ gives a deformation of .Ho;m WD fn0;kgk�1/ as a weak A1–algebra.

For a dg Lie algebra, there is the notion of gauge transformation. A gauge transfor-mation defines an equivalence relation � between elements in L; two elements inL are equivalent iff they are related by a gauge transformation. In particular, gaugetransformations preserves the solution space MC.L/. Thus, the quotient space ofMC.L/ by the equivalence relation � is well-defined:

M.L/ WDMC.L/=� :The moduli space of deformations is defined as this M.L/. In particular, one has iso-morphic spaces M.L/ for any L of the same L1–homotopy type. Thus, deformationtheory is in general controlled by an L1 homotopy class of a dg Lie algebra.

In general, to construct or even show the existence of an L1–morphism is a verydifficult problem. However, there exists a pair of a dg Lie algebra and an L1–algebrawhere the existence of an L1–morphism between them is guaranteed in some sense.We shall explain this below.

6 Homotopy algebra and string theory

Let us start from a ‘definition’ of string theory which provides the main motivationfor obtaining a pair of a dg Lie algebra and an L1–algebra together with an L1–morphism between them.

A string is a one dimensional object whose ‘trajectory’ (worldsheet) is described asa Riemann surface (two-dimensional object). In string theory, the most fundamental

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quantities are the scattering amplitudes of strings. The most general Riemann sur-faces to be concerned with are those with genera, boundaries, and punctures on theboundaries and/or in the interior, where a puncture on a boundary (resp. in the interior)corresponds to an open (resp. closed) string insertion. For a fixed appropriate fieldtheory on Riemann surfaces, the scattering amplitudes are obtained by choosing asuitable compactification of the moduli spaces of punctured Riemann surfaces; thescattering amplitudes are integrals over the compactified moduli spaces. The collectionof the scattering amplitudes obtained as above are endowed with special algebraicstructures, associated to the stratifications of the compactifications of the moduli spaces.In this sense, a definition of a string theory is the pair of an operad (associated to thecompactified moduli spaces) and a representation of the operad (an algebra over theoperad).

In particular, if we consider a sigma-model on a Riemann surface, ie, a field theorywhose fields are maps from the Riemann surfaces to a target space M , the representationobtained by the field theory has some information about the geometry of M .

As we explained, an A1–algebra .A;m/ is obtained as a representation of the A1–operad A1 on A. So a deformation of .A;m/ is a deformation of a representation ofthe A1–operad A1 on the fixed graded vector space A.

Here, recall that the A1–operad A1 is a structure which is associated to the realcompactification of the moduli spaces of disks with punctures on the boundaries.

On the other hand, a representation of the L1–operad L1 on a fixed graded vectorspace L is an L1–algebra .L; l/.

Now, if we fix a tree open string theory and a tree closed string theory on Riemannsurfaces, we obtain an A1–algebra .Ho;m/ and the dg Lie algebra .Coder.T c.Ho//;

D; Œ ; �/ controlling its deformation for the tree open string theory, and also an L1–algebra .Hc ; l/ for the tree closed string. Moreover, the tree open–closed string systemprovides a representation of the OC1–operad on H WDHc ˚Ho , that is, an OCHA.H; l; n/. Together with Theorem 5.3, by considering an appropriate tree open–closedstring (field) theory, one can get a pair of a dg Lie algebra and an L1–algebra togetherwith a non-trivial L1–morphism betweeen them.

One example of this is Kontsevich’s set-up [42] for deformation quantization [3; 4](see [38]). Furthermore, as L1–algebras Hc of closed string (field) theories, onecan consider for instance the dg Lie algebra controlling the extended deformation ofcomplex structures [2] (which is what is called the B–model in string theory (cf [38])),or the dg Lie algebra controlling deformation of generalized complex structure [22],etc.

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Then, OCHAs should guarantee the existence of corresponding deformations of theA1–structures. Currently, it should be a very interesting problem to descibe explicitlysuch a deformation of an A1–structure as was done in the case of �–product fordeformation quantization [42]. Some attempts in this direction can be found in [31] forB–twisted topological strings and [51] for the case of generalized complex structures.

In these situations, which are related to mirror symmetry, the L1–algebra correspond-ing to the tree closed string theory is (homotopy equivalent to) trivial, which impliesthat the deformation of the corresponding A1–structure is unobstructed (see [38]). Insuch situations, deformation of A1–structures can be thought of as a (homological)algebraic description of an L1–algebra .Hc ; l/ describing deformation of a geometry(see homological mirror symmetry by Kontsevich [41]).

Acknowledgments In working on OCHA, we had had valuable discussions withE Harrelson, A Kato, T Kimura, T Lada, M Markl, S A Merkulov, H Ohta, A Voronovand B Zwiebach.

JS is supported in part by NSF grant FRG DMS-0139799 and US–Czech Republicgrant INT-0203119.

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Research Institute for Mathematical Sciences, Kyoto UniversityKyoto 606-8502, Japan

Department of Mathematics, University of PennsylvaniaPhiladelphia, PA 19104-6395, USA

kajiura@kurims.kyoto-u.ac.jp, jds@math.upenn.edu

Received: 31 May 2006 Revised: 22 January 2007

Geometry & Topology Monographs, Volume 13 (2008)